Rao's score, Neyman's C(α) and Silvey's LM tests: An essay on historical developments and some new results

Rao's (Proc. Cambridge Philos. Soc. 44 (1948a) 50) seminal paper introduced a fundamental principle of testing based on the score function as an alternative to likelihood ratio and Wald tests. Neyman's (In: Grenander, (Ed.), Probability and Statistics, the Harald Cramér Volume, Almqvist and Wiksell, Uppsala, pp. 213-234) approach, in view of the presence of nuisance parameters, emphasized the generality and attractive features of the score-based tests. Silvey (Ann. Math. Statist. 30 (1959) 389) rediscovered the score test as a Lagrange multiplier test. Breusch and Pagan's (Rev. Econom. Stud. 47 (1980) 239) exposition of the score test in a general framework in the context of econometric modeling resulted in an increased activity on specification testing in econometrics. In this paper we trace these historical developments emphasizing the optimality features of tests based on scores and their usefulness in practical problems in statistics and econometrics. In so doing we give some new results, present easier computation of score-based tests and alternative derivations of some known results. We also discuss a connection between Rao's score test and the seemingly unrelated literature of Fisher's discriminant function, Mahalanobis' D² and Hotelling's T².